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Conformal Mappings and 3D Imaging

Conformal Mappings and 3D Imaging. Ronald Lok Ming LUI Department of Mathematics, The Chinese University of Hong Kong. Math 3310 Supplementary Introduction. Advances in 3D acquisition techniques (such as 3D laser scanning, structured light scanning);

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Conformal Mappings and 3D Imaging

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  1. Conformal Mappings and 3D Imaging Ronald Lok Ming LUIDepartment of Mathematics, The Chinese University of Hong Kong Math 3310Supplementary Introduction

  2. Advances in 3D acquisition techniques (such as 3D laser scanning, structured light scanning); • Real world objects can be captured effectively; • 3D data are represented by Point Clouds, Meshes or Implicit Representation (Level set). Motivation: 3D Imaging Structured light scanning technology (from Gu’s group)

  3. Question: • How to connect 2D Imaging and 3D Imaging? • Goal: • See how conformal mappings can • help! Surface decoration Motivation: 3D Imaging Surface matching Bad triangulation Noisy data Surface holes

  4. Brain Conformal Parameterization: A canonical domain for brain surface analysis! Application: Brain Registration Genus 0 Open surface (disk is removed at the back)

  5. Goal: Solve equations on the surface by mapping it onto the 2D conformal parameter domain. • Differential operators are computed on 2D domain with simple formula. • Example: Solving PDEs on surfaces: Conformal Approach Projected PDE on conformal domain take simple expression! Conformal factor

  6. Applications: Imaging on surfaces

  7. Surface denoising: Applications: Imaging on surfaces Original Noisy surface Denoised surface

  8. Results: Surface inpainting using conformal factor Incomplete surface Inpainting minimizing (Gradient of normals) Our result

  9. Results: Surface inpainting using conformal factor

  10. Texture mapping = map image onto a surface (for surface decoration etc) Idea: 1. Map vertices to 2D positions of an image; (Correspondence guided by landmark features) 2. Color value is assigned for each vertex; 3. Color value inside the face by linear interpolation. Constrained Texture Mapping 2D Image Textured surface mesh

  11. Optimized conformal mappings Constrained Texture Mapping

  12. Optimized conformal mapping Constrained Texture Mapping

  13. What have you learned from Math 3310? • Main Goal: Solving real world problem! • How? • Convert the problem to math. Equations (e.g. PDEs) • (Chapter 1) Recap: Math 3310 • How to solve the Math. Eqt./PDEs? • 1. Analytic way: Find the exact solution/ simplify the PDE to simple PDE for which exact solution can be found. • (Commonly used techniques in applied maths: • Spectral method: Fourier, Laplace etc…) (Chapter 2)

  14. What have you learned from Math 3310? (Final exam starts from discrete!) • How to solve the Math. Eqt./PDEs? • 2. Discrete way: • - Discrete Fourier transform: approximate the N Fourier coefficient to approximate the Fourier series solution.(Chapter 2 Supplementary) • - Discretize the domain/eqts: solving linear systems! Recap: Math 3310

  15. What have you learned from Math 3310? • How to solve Big Linear Systems? • Iterative method • - Splitting method: (Chapter 3) Jacobi, G-S, SOR • A = N – P. • (Convergence? Best parameters for SOR?) • - Descent method: (Chapter 3 Supplementary) • Steepest gradient descent: flow along gradient • Conjugate gradient method: descent directions are • conjugate Recap: Math 3310

  16. What have you learned from Math 3310? • How t0 find spectral radius? • 1. Power method • 2. Simultaneous iteration (power method on matrix) • 3. QR method • 4. Inverse power method • 5. Inverse power method with shift • 6. Rayleigh Quotient iteration Recap: Math 3310 ALL SIX METHODS COME FROM POWER METHOD!!

  17. What have you learned from Math 3310? • How to solve eqts on complicated domain? • Conformal mapping! Recap: Math 3310

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